Problem 6 - Algebra 1

Question

Does the equation model direct variation, inverse variation, or neither? $$x=\frac{4}{y}$$

Step-by-Step Solution

Answer

The equation \(x=\frac{4}{y}\) models an inverse variation.

Step 1: Identify the Form of the Given Equation

In the equation \(x=\frac{4}{y}\), it can be rewritten in a form where the product of two variables equals a constant. Multiply both sides by y to get \(xy=4\). Here, the constant is 4.

Step 2: Analyze the Form of the Equation

The equation \(xy=4\) is in the form of an inverse variation, where the product of \(x\) and \(y\) is a constant. This is not in the form \(y=kx\) which is a direct variation.

Step 3: Determining the Type of Variation

As the equation \(xy=4\) follows the inverse variation and not the direct variation, the given equation represents an inverse variation.

Key Concepts

Understanding Direct Variation

In mathematics, direct variation refers to a relationship between two variables where one is a constant multiple of the other. The formula for direct variation is given as \( y = kx \), where \( k \) is the constant of variation. This means that as one variable increases, the other variable also increases proportional to the constant \( k \).
The key idea is that any changes in one variable result in consistent and predictable changes in the other, maintaining a direct relationship.

• If you double the value of \( x \), the value of \( y \) also doubles when \( y = kx \).
• This linearity is characterized by a straight line through the origin in a graph.

Direct variation is commonly found in problems involving speeds, costs, or any scenario where two quantities are directly proportional to one another.

Exploring the Product of Variables

The product of variables is a crucial concept in understanding different types of mathematical variation. It plays a significant role in distinguishing between direct and inverse variation. In the context of variations, when you have an equation like \( xy = c \), where \( c \) is a constant, it indicates that the product of \( x \) and \( y \) remains the same. This constancy of the product is the hallmark of inverse variation, not direct variation.

• For example, if \( x \) increases, then \( y \) must decrease to keep \( xy \) equal to \( c \).
• This relationship leads to a non-linear graph, often hyperbolic in nature.

Unlike direct variation, where the output varies linearly, products in these equations fluctuate in opposite directions to maintain balance.

Identifying Variation Types

Identifying whether an equation models direct or inverse variation involves examining the form and relationships of the variables involved. To determine the type of variation, consider the forms:

• Direct variation takes the linear form \( y = kx \).
• Inverse variation is represented as \( xy = c \).

If an equation is in the form \( y = kx \), it's classified as direct variation because one variable is proportional to another. Conversely, if it matches \( xy = c \), it signifies inverse variation, illustrating that the variables have a multiplicative relationship that equals a constant.
In the exercise provided, the equation \( x = \frac{4}{y} \) was simplified to \( xy = 4 \). This shows that \( x \) and \( y \) multiplied give a constant, confirming inverse variation, as opposed to direct.